An Algorithmic Approach for Stability of an Autonomous System

Many phenomena in biology can be modeled as a system of first order differential equations x = ax + by, y=cx+dy. An example of such a system is the prey-predator model. To interpret the results we have to obtain full information on the system of equations such as the stability of the equilibrium points of the system. This requires in depth knowledge of differential equations. The literature often emphasizes on the analytical methods to obtain results regarding the stability of the equilibrium points. This is possible to achieve for small systems such as a 2 x 2 system. The non-mathematician researchers often do not have the analytical tools to understand the model fully. Very often what they are interested in is the information regarding the critical points and their stability without going through the tedious mathematical analysis. This calls for user friendly tools for the non-mathematicians to use in order to answer their problem at hand. The objective of this research is to establish an algorithm to determine the stability of a more general system. By doing so we will be able to help those who are not familiar with analytical methods to establish stability of systems at hand The following algorithm is' employed in developing the software: L 1. Search for critical point is conducted. L2. Eigenvalues of the linear system are computed. These values are obtained from the characteristic equation IA - All = 0 , where A. is an eigenvalue and F or the nonlinear system, linearization process around the critical points are carried out. L3. Stability of system is determined. L4. Trajectory of the system is plotted in the phase plane. To develop the software we use the C programming language. It is hoped that the software developed will be of help to researchers in the field of mathematical biology to understand the concept of stability in their model.

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